Asymptotic Bounds for t(3,n) and an Application to t(4,n)

A set of vertices $X\subseteq V$ in a simple graph $G(V,E)$ is irredundant if each vertex $x\in X$ is either isolated in the induced subgraph $G[X]$ or else has a private neighbor $y\in V\setminus X$ that is adjacent to $x$ and to no other vertex of $X$. The \emph{mixed Ramsey number} $t(m,n)$ is the smallest $N$ for which every red-blue coloring of the edges of $K_N$ has an $m$-element irredundant set in the blue subgraph or an $n$-element independent set in the red subgraph. The irredundant Ramsey number $s(m,n)$ is the smallest $N$ for which every red-blue coloring of the edges of $K_N$ has an $m$-element irredundant set in the blue subgraph or an $n$-element irredundant set in the blue subgraph. In this paper, we determine $t(3,n)$ and $s(3,n)$ up to a constant factor by showing that $t(3,n)=O\left(n^{5/4}/{\log{n}}\right)$, which improved the best upper bound due to Rousseau and Speed in [Comb. Probab. Comput. 12 (2003), 653-660]. As an application, we verify a conjecture for $m=4$ proposed by Chen, Hattingh, and Rousseau in [J. Graph Theory 17(2) (1993), 193-206].